$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ #[Chapter 9. Wave equation](id:chapter-9) ##[Wave equation in dimensions $3$ and $2$](id:sect-9.1) > 1. [$3D$-Wave equation: special case](#sect-9.1.1) > 2. [$3D$-Wave equation: general case](#sect-9.1.2) > 3. [Spherical means](#sect-9.1.3) > 4. [$2D$-wave equation: method of descent](#sect-9.1.4) > 5. [Limiting amplitude principle](#sect-9.1.5) > 6. [Remarks](#sect-9.1.6) ###[$3D$-Wave equation: special case](id:sect-9.1.1) Consider Cauchy problem for $3$-dimensional wave equation \begin{align} & u\_{tt}-c^2\Delta u=f,\label{eq-9.1.1}\\\\[3pt] & u|\_{t=0}=g,\label{eq-9.1.2}\\\\[3pt] & u\_t|\_{t=0}=h.\label{eq-9.1.3} \end{align} Assume first that $f=g=0$. We claim that in this case as $t\>0$ \begin{equation} u(x,t)= \frac{1}{4\pi c ^2 t} \iint \_{S(x,ct)} h(y)\,d\sigma \label{eq-9.1.4} \end{equation} where we integrate along sphere $S(x,ct)$ with a center at $x$ and radius $ct$; $d\sigma$ is an area element. Let us prove (\ref{eq-9.1.4}) first as $h(x)=e^{ix\cdot \xi}$ with $\xi\in \mathbb{R}^3\setminus 0$; we use the standard notation $x\cdot \xi=x\_1\xi\_1+x\_2\xi\_2+x\_3\xi\_3$. In this case \begin{equation} u(x,t)=e^{ix\cdot \xi}c^{-1}|\xi|^{-1}\sin (ct|\xi|) \label{eq-9.1.5} \end{equation} is obviously a solution to Cauchy problem (\ref{eq-9.1.1})--(\ref{eq-9.1.3}). On the other hand, the right-hand expression of (\ref{eq-9.1.4}) becomes \begin{equation\*} \frac{1}{4\pi c ^2 t}\iint \_{S(x,ct)} e^{iy\cdot \xi}\,d\sigma= \frac{1}{4\pi c ^2 t}e^{ix\cdot \xi}\iint \_{S(0,ct)} e^{iz\cdot \xi}\,d\sigma \end{equation\*} where we changed variable $y=x+z$ with $z$ running $S(0,ct)$ (sphere with a center at $0$) and we need to calculate integral in the right-hand expression. Let us select coordinate system in which $\xi=(0,0,\omega)$ with $\omega =|\xi|$ and introduce corresponding spherical coordinates $(\rho,\phi,\theta)$; then on $S(0,ct)$ $\rho=ct$, $z\cdot \xi= z\_3\omega= ct \omega \cos(\phi)$ and $d\sigma=c^2t^2 \sin(\phi)d\phi d\theta$; so integral becomes \begin{multline\*} c^2t^2 \int\_0^\pi e^{i ct \omega \cos(\phi)}\sin(\phi) \,d\phi \int\_0^{2\pi} d\theta=\\\\ -2\pi c^2t^2 \int\_0^\pi e^{i ct \omega \cos(\phi)} \,d\cos(\phi) = 2\pi c^2t^2 \frac{1}{ic t\omega} \Bigl( e^{i ct \omega }- e^{-i ct \omega } \Bigr)=\\\\ 4\pi ct \omega^{-1} \sin (ct\omega) \end{multline\*} and multiplying by $e^{ix\cdot\xi}$ and dividing by $4\pi c^2t$ we get $e^{ix\cdot\xi} c^{-1}|\xi|^{-1} \sin (ct|\xi|)$ which is the right-hand expression of (\ref{eq-9.1.5}). So, for $h(x)=e^{ix\cdot \xi}$ (\ref{eq-9.1.4}) has been proven. However the general function $h(x)$ could be decomposed into such special functions using *multidimensional Fourier transform* and *multidimensional Fourier integral* which is nothing but repeated $1$-dimensional Fourier transform and Fourier integral: \begin{gather} h(x)=\iiint \hat{h}(\xi) e^{ix\cdot\xi}\,d\xi,\label{eq-9.1.6}\\\\[3pt] \hat{h}(\xi)=(2\pi)^{-n}\iiint h(x) e^{-ix\cdot\xi}\,dx\label{eq-9.1.7} \end{gather} and therefore (\ref{eq-9.1.4}) extends to general functions as well. **[Remark 1.](id:remark-9.1.1)** We should deal with the fact that only decaying functions could be decomposed into Fourier integral, but this is easy due to the fact that integral in (\ref{eq-9.1.4}) is taken over bounded domain. ###[$3D$-Wave equation: general case](id:sect-9.1.2) To cover $t\<0$ we replace (\ref{eq-9.1.4}) by \begin{equation} u(x,t)= \frac{1}{4\pi c t} \iint \_{S(x,c|t|)} h(y)\,d\sigma \label{eq-9.1.8} \end{equation} which is obvious as $u$ must be odd with respect to $t$. Consider now $g\ne 0$. Let $v$ be given by (\ref{eq-9.1.8}) with $h$ replaced by $h$; then \begin{align\*} & v\_{tt}-c^2\Delta v=0,\\\\[3pt] & v|\_{t=0}=0,\\\\[3pt] & v\_t|\_{t=0}=g. \end{align\*} Then $\Delta v|\_{t=0}=0$ and therefore $v\_{tt}|\_{t=0}=0$ and therefore differentiating equation with respect to $t$ we conclude that $u:=v\_t$ solves \begin{align\*} & u\_{tt}-c^2\Delta u=0,\\\\[3pt] & u|\_{t=0}=g,\\\\[3pt] & u\_t|\_{t=0}=0. \end{align\*} Now \begin{equation} u(x,t)= \frac{\partial\ }{\partial t} \Bigl(\frac{1}{4\pi c^2 t} \iint \_{S(x,c|t|)} g(y)\,d\sigma\Bigr). \label{eq-9.1.9} \end{equation} Therefore solving separately (\ref{eq-9.1.1})--(\ref{eq-9.1.3}) for $f=g=0$ (given by (\ref{eq-9.1.8})) and for $f=h=0$ (given by (\ref{eq-9.1.9}) and adding solutions due to linearity we arrive to \begin{equation} u(x,t)= \frac{\partial\ }{\partial t} \Bigl(\frac{1}{4\pi c^2 t} \iint\_{S(x,c|t|)} g(y)\,d\sigma\Bigr)+ \frac{1}{4\pi c ^2 t} \iint \_{S(x,c|t|)} h(y)\,d\sigma \label{eq-9.1.10} \end{equation} covering case $f=0$. To cover case of arbitrary $f$ but $g=h=0$ we apply Duhanel integral formula (see [Subsection 2.5.2](../Chapter2/S2.5.html#sect-2.5.2)). Consider problem \begin{align\*} & U\_{tt}-c^2\Delta U=0,\\\\[3pt] & U|\_{t=\tau}=0,\\\\[3pt] & U\_t|\_{t=\tau}=f(x,\tau). \end{align\*} Its solution is given by \begin{equation\*} U(x,t,\tau) = \frac{1}{4\pi c^2(t-\tau)} \iint\_{S(x,c|t-\tau|) } f(y,\tau)\,d\sigma \end{equation\*} and therefore \begin{equation} u(x,t) = \int\_0^t\frac{1}{4\pi c^2(t-\tau)} \iint\_{S(x,c|t-\tau|)} f(y,\tau)\,d\sigma d\tau. \label{eq-9.1.11} \end{equation} Assembling (\ref{eq-9.1.10}) and (\ref{eq-9.1.11}) together we arrive to *Kirchhoff formula* \begin{multline} u(x,t)= \frac{\partial\ }{\partial t} \Bigl(\frac{1}{4\pi c^2 t} \iint \_{S(x,c|t|)} g(y)\,d\sigma\Bigr)+ \frac{1}{4\pi c^2 t} \iint \_{S(x,c|t|)} h(y)\,d\sigma +\\\\ \int\_0^t \frac{1}{4\pi c^2(t-\tau)} \iint\_{S(x,c|t-\tau|) } f(y,\tau)\,d\sigma d\tau. \qquad \label{eq-9.1.12} \end{multline} providing solution to (\ref{eq-9.1.1})--(\ref{eq-9.1.3}). **[Remark 2.](id:remark-9.1.2)** As $t\>0$ one can rewrite the right-hand expression in (\ref{eq-9.1.11}) as \begin{equation} \iiint\_{B(x,ct)} \frac{1}{4\pi c^2 |x-y|}f(y,t-c^{-1}|x-y|)\,dy \label{eq-9.1.13} \end{equation} where we we integrate over ball $B(x,ct)$ of radius $ct$ with the center at $x$. ###[Spherical means](id:sect-9.1.3) **[Definition 1.](id:definition-9.1.1)** $M\_r(h, x)=\frac{1}{4\pi r^2} \int\_{S(x,r)} h(y)\,d\sigma$ is a *spherical mean* of $h$. Recall that $4\pi r^2$ is an area of $S(x,r)$. Therefore (\ref{eq-9.1.8}) is exactly $ u(x,t)= t M\_{c|t|} (h,x)$ and all other formulae (\ref{eq-9.1.9})--(\ref{eq-9.1.13}) could be modified similarly. ###[$2D$-wave equation: method of descent](id:sect-9.1.4) Consider now the same problem (\ref{eq-9.1.1})--(\ref{eq-9.1.3}) but in dimension $2$. To apply (\ref{eq-9.1.12}) we introduce a third spatial variable $x\_3$ and take $f$, $g$, $h$ not depending on it; then $u$ also does not depend on $x\_3$ and solves original $2D$-problem. So, the right-hand expression in (\ref{eq-9.1.8}) becomes for $\pm t\>0$ \begin{equation} \frac{1}{4\pi c^2t} \iint \_{S(x,c|t|)} h(y)\,d\sigma= \pm \frac{1}{2\pi c} \iint \_{B(x,c|t|)} \frac{h(y)}{\sqrt{c^2t^2-|x-y|^2}}\,dy \label{eq-9.1.14} \end{equation} where $y=(y\_1,y\_2)$ and we took into account that $S(x,ct)$ covers disk $B(x,c|t|)$ twice (so factor $2$ appears) and $d\sigma = \pm \frac{ct}{\sqrt{c^2t^2-|x-y|^2}}\,dy$, $dy=dy\_1dy\_2$. Thus (\ref{eq-9.1.12}) implies that for $\pm t\>0$ \begin{multline} u(x,t)= \pm\frac{\partial\ }{\partial t} \Bigl(\frac{1}{2\pi c} \iint \_{B(x,c|t|)} \frac{g(y)}{\sqrt{c^2t^2-|x-y|^2}} \,d y\Bigr) \\\\ \pm \frac{1}{2\pi c } \iint \_{B(x,c|t|)} \frac{h(y)}{\sqrt{c^2t^2-|x-y|^2}}\,dy \\\\ \pm \int\_0^t \frac{1}{4\pi c } \iint\_{B(x,c|t-\tau|) } \frac{f(y,\tau)}{\sqrt{c^2(t-\tau)^2-|x-y|^2}}\,dy d\tau. \qquad \label{eq-9.1.15} \end{multline} ###[Limiting amplitude principle](id:sect-9.1.5) Let $n=3$. Consider solution to inhomogeneous wave equation with a special right-hand expression \begin{equation} \Delta u -c^{-2}u\_{tt} = f(x)e^{i\omega t} \label{eq-9.1.16} \end{equation} where $\omega\ne 0$ and $f(x)$ does not depend on $t$ and fast decays as $|x|\to \infty$. Assume that $g(x)=u(x,0)$ and $h(x)=u\_t (x,0)$ also fast decay as $|x|\to \infty$. Plugging all these functions into Kirchhoff formula (\ref{eq-9.1.12}) and considering $|t|\gg 1$ and fixed $x$ we see that \begin{equation} |u (x,t) - v^\pm \_\omega (x)e^{i\omega t}|\to 0 \qquad \text{as\ \ }t\to \pm \infty \label{eq-9.1.17} \end{equation} with \begin{equation} v^\pm\_\omega= -\frac{1}{4\pi}\iiint |x-y|^{-1}e^{\mp i\omega c^{-1}|x-y|}\, dy. \label{eq-9.1.18} \end{equation} One can check easily that $v=v^\pm\_\omega$ satisfies *Helmholtz equation* \begin{equation} \bigl(\Delta +\frac{\omega^2}{c^2}\bigr)v = f(x) \label{eq-9.1.19} \end{equation} with *Sommerfeld radiating conditions* \begin{align} &v = o(1) &&\text{as }r\to \infty,\label{eq-9.1.20}\\\\ &( \partial\_r \mp ic^{-1}\omega )v= o( r^{-1}) &&\text{as }r\to \infty \label{eq-9.1.21} \end{align} where $r:=|x|$ and $\partial\_r := |x|^{-1} x\cdot \nabla$. This is called *Limiting amplitude principle* **[Remark 3.](id:remark-9.1.3)** a. Formula (\ref{eq-9.1.18}) gives us Green function for problem (\ref{eq-9.1.19})-(\ref{eq-9.1.21}) \begin{equation} G^\pm\_\omega (x,y)= -\frac{1}{4\pi} |x-y|^{-1}e^{\mp i\omega c^{-1}|x-y|} \label{eq-9.1.22} \end{equation} which coincides as $\omega=0$ with a Green function $G(x,y)=-\frac{1}{4\pi} |x-y|^{-1}$; b. However now we have two Green functions as (\ref{eq-9.1.22}) distinguishes and between them and $u(x,t)$ has different amplitudes $v^\pm\_\omega$ as $tto \pm \infty$. c. For fast–decaying $f$ one can replace in (\ref{eq-9.1.20}) and (\ref{eq-9.1.21}) $o(1)$ and $o(r^{-1})$ by $O(r^{-1})$ and $O(r^{-2})$ respectively. ###[Remarks](id:sect-9.1.6) **[Remark 4.](id:remark-9.1.4)** Formulae (\ref{eq-9.1.13}) and (\ref{eq-9.1.15}) could be generalized to the case of odd $n\ge 3$ and even $n\ge 2$ respectively. These formulae imply that $u(x,t)$ does not depend on $g(y),h(y)$ with $|y-x|\>ct$ and on $f(y,\tau)$ with $|y-x|\>c|t-\tau|$. This could be interpreted as "nothing propagates with a speed exceeding $c$". We will prove it again by completely different method in the next [Section 9.2](./S9.2.html). **[Remark 5.](id:remark-9.1.5)** a. As $n\ge 3$ is odd $u(x,t)$ is given by the following formula \begin{align} u(x,t)= c^{1-n}\kappa\_n \frac{\partial\ }{\partial t} &\left(\frac{1}{t}\frac{\partial\ }{\partial t}\right)^{\frac{n-3}{2}} \Biggl(t^{-1}\iint\_{S(x,c|t|)}g(y)\,d \sigma\Biggr)\notag\\\\ +c^{1-n}\kappa\_n &\left(\frac{1}{t}\frac{\partial\ }{\partial t}\right)^{\frac{n-3}{2}} \Biggl(t^{-1}\iint\_{S(x,c|t|)}h(y)\,d \sigma\Biggr) \label{eq-9.1.23} \end{align} provided $f=0$ (which could be generalized to $f\ne 0$ using Duhamel principle). b. As $n\ge 2$ is even $u(x,t)$ is given by the following formula obtained by the method of descent \begin{align} \hskip{-20pt}u(x,t)\qquad&\notag\\\\= c^{1-n}\kappa\_n \frac{\partial\ }{\partial t} &\left(\frac{1}{t}\frac{\partial\ }{\partial t}\right)^{\frac{n-2}{2}} \Biggl(\iiint\_{B(x,c|t|)}\frac{g(y)}{(c^2t^2-|x-y|^2)^{\frac{1}{2}}}\,dy \Biggr)\notag\\\\ +c^{1-n}\kappa\_n &\left(\frac{1}{t}\frac{\partial\ }{\partial t}\right)^{\frac{n-2}{2}} \Biggl(\iiint\_{B(x,c|t|)}\frac{g(y)}{(c^2t^2-|x-y|^2)^{\frac{1}{2}}}\,dy \Biggr) \label{eq-9.1.24} \end{align} provided $f=0$ (which could be generalized to $f\ne 0$ using Duhamel principle). c. Here $\kappa\_n$ is a numerical coefficient which could be easily calculated from $g \equiv 1,\ h\equiv 0,\ f\equiv 0 \implies u\equiv 1$. c. In particular, for odd $n\ge 3$ solution $u(x,t)$ does not depend on $g(y),h(y)$ with $|y-x|\< ct$ and on $f(y,\tau)$ with $|y-x|\< c|t-\tau|$. This could be interpreted as "no leftovers after front passed with a speed $c$". In mathematical literature this is called *Huygens principle* (there is another Huygens principle aka [*Huygens-Fresnel principle*](http://en.wikipedia.org/wiki/Huygens%E2%80%93Fresnel_principle)). This property is a rare commodity: adding lower-order terms to the equation breaks it. --------------- [$\Leftarrow$](../Chapter8/S8.A.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](./S9.2.html)